Finite Group Databasebeta

← #27158510

Subgroups — #27158510

Order 12416.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261814C211yesmaximal, characteristic, complemented
2#20261813C211yesmaximal, characteristic, complemented
3#20261782C212yesmaximal, complemented
4#20261780C212yesmaximal, complemented
5#20261778C212yesmaximal, complemented
6#20261777C212yesmaximal, complemented
7#20261750C211yesmaximal, characteristic, complemented
8D97⋊#104C211yesmaximal, characteristic, complemented
9C2×#15690874C211yesmaximal, characteristic, complemented
10M16×#87604C211yesmaximal, characteristic, complemented
11#20261531C211yesmaximal, characteristic
12#15690898C2212yescomplemented
13#15690897—21no—
14#15690897—21no—
15#15690897—22no—
16#15690897C2212yescomplemented
17#15690897C2212yescomplemented
18#15690897C2212yescomplemented
19C2×#13095521—21no—
20C2×#13095521C2211yescharacteristic
21D97⋊M16—21no—
22D97⋊M16C2211yescharacteristic
23D97⋊M16C2211yescharacteristic
24C2×#13095520—22no—
25C2×C4×#87604C2211yescharacteristic
26#15690881C2212yescomplemented
27#15690880C2212yescomplemented
28#15690875C2211yescharacteristic, complemented
29#15690875C2211yescharacteristic, complemented
30#15690874C2212yescomplemented
31#15690874C2212yescomplemented
32#15690874C2211yescharacteristic, complemented
33#15690874C2211yescharacteristic, complemented
34#15690864C2212yescomplemented
35C8×#87604C2212yescomplemented
36C2×#13095510C2211yescharacteristic
37#41×D97C2211yescharacteristic
38#15690843C411yescharacteristic, complemented
39#15690840C411yescharacteristic, complemented
40#15690839C412yescomplemented
41#15690836C411yescharacteristic, complemented
42#15690834C412yescomplemented
43#15690831C2212yes—
44#15690829C2212yes—
45M16×D97C2211yescharacteristic
46AGL(1,ℤ/8)×C97C411yescharacteristic, complemented
47C22×#87604—21no—
48#13095521—21no—
49#13095521—22no—
50#13095521D411yescharacteristic
51#13095521D411yescharacteristic
52#13095521C2312yes—
53#13095521C2311yescharacteristic
54#13095521C2311yescharacteristic
55#13095520—21no—
56#13095520—22no—
57#13095520—22no—
58#13095520—22no—
59#13095520D412yes—
60#13095519—21no—
61#13095519—22no—
62#13095519C2312yes—
63C2×#11715897—21no—
64C2×#11715897—22no—
65C2×#11715897—22no—
66C2×#11715897—22no—
67C4×#87604—21no—
68C4×#87604C2311yescharacteristic
69C4×#87604C2311yescharacteristic
70#13095512C2311yescharacteristic
71#13095511—21no—
72#13095511C2×C411yescharacteristic
73#13095510—21no—
74#13095510—21no—
75#13095510—21no—
76#13095510C2×C412yes—
77#13095510C2311yescharacteristic
78#13095510C2312yes—
79C2×Dic194C2×C411yescharacteristic
80Q8×D97C2×C411yescharacteristic
81C2×C22⋊D97—21no—
82D4×D97—21no—
83D4×D97C2×C411yescharacteristic
84C22×Dic97—21no—
85C2×C4×D97—21no—
86C2×C4×D97C2311yescharacteristic
87Dic388C2×C412yescomplemented
88#13095501C2×C412yescomplemented
89#13095500C2×C412yescomplemented
90#13095499C2×C412yescomplemented
91#13095499C2×C412yescomplemented
92#13095496C2×C412yescomplemented
93#13095495C2×C411yescharacteristic
94#13095494C2×C412yes—
95C8×D97C2312yes—
96#41×C97C2×C411yescharacteristic
97C2×D4×C97C2×C411yescharacteristic
98SD16×C97C2×C412yescomplemented
99D8×C97C2×C412yescomplemented
100M16×C97C2×C411yescharacteristic
101C23×C97—21no—
102C22×D97—21no—
103C22×D97#4111yescharacteristic
104C2×#87604—21no—
105C2×#87604—21no—
106C2×#87604—42no—
107Q8×C97C22×C411yescharacteristic
108D4×C97—21no—
109D4×C97—21no—
110D4×C97C22×C411yescharacteristic
111D4×C97C22×C412yes—
112C2×C4×C97—21no—
113C2×C4×C97C22×C411yescharacteristic
114C22⋊D97—21no—
115C22⋊D97—22no—
116C22⋊D97—21no—
117C22⋊D97—22no—
118C22⋊D97—21no—
119C2×Dic97—21no—
120C2×Dic97—22no—
121C2×Dic97—21no—
122C2×Dic97—22no—
123C2×Dic97C2×D411yescharacteristic
124D388—21no—
125D388C22×C411yescharacteristic
126C4×D97—21no—
127C4×D97—21no—
128C4×D97—21no—
129C4×D97C22×C411yescharacteristic
130C4×D97C2×D411yescharacteristic
131C4×D97#4111yescharacteristic
132C4×D97C2411yescharacteristic
133Dic194—21no—
134Dic194—21no—
135Dic194C22×C411yescharacteristic
136Dic194C22×C412yes—
137#11715897—21no—
138#11715897—21no—
139#11715897—22no—
140#11715897C2×D412yes—
141#11715897C2×D412yes—
142C8×C97C22×C412yes—
143#11715895C22×C412yes—
144C22×C97—22no—
145C22×C97—21no—
146C22×C97—81no—
147C22×C97C4×D411yescharacteristic
148C2×D97—21no—
149C2×D97—41no—
150C2×D97—81no—
151C2×D97C4×D411yescharacteristic
152C2×D97C2×#4111yescharacteristic
153#87604—22no—
154#87604—41no—
155C4×C97—21no—
156C4×C97C4×D411yescharacteristic
157C4×C97C23×C411yescharacteristic
158Dic97—21no—
159Dic97—22no—
160Dic97C4×D411yescharacteristic
161Dic97C22×D411yescharacteristic
162C2×C97—21no—
163C2×C97—41no—
164C2×C97—42no—
165C2×C97C2×C4×D411yescharacteristic
166D97—21no—
167D97—21no—
168D97—41no—
169#2956—971nomaximal, Sylow
170C97#295611yescharacteristic, complemented, Sylow
171#575—971no—
172#568—971no—
173#568—971no—
174#444—972no—
175#444—972no—
176#443—972no—
177#443—972no—
178#430—971no—
179C2×#104—971no—
180C2×#104—971no—
181C4×M16—971no—
182C2×#41—971no—
183C2×#41—971no—
184#137—971no—
185#137—972no—
186#137—971no—
187AGL(1,ℤ/8)—971no—
188AGL(1,ℤ/8)—972no—
189AGL(1,ℤ/8)#8760411yescharacteristic, complemented
190#135—972no—
191#135—972no—
192#131—972no—
193#131—972no—
194#131—972no—
195#131—972no—
196#131—1941no—
197#131—1941no—
198#131—1942no—
199C2×M16—971no—
200C2×M16—971no—
201C2×M16—971no—
202C2×M16—971no—
203C2×M16—1942no—
204C2×M16—1941no—
205C2×M16—1941no—
206C2×C42—971no—
207#108—972no—
208#108—972no—
209#104—972no—
210#104—971no—
211#104—972no—
212#104—971no—
213#104—971no—
214#104—971no—
215#97—972no—
216C4×C8—972no—
217#41—971no—
218#41—972no—
219#41—972no—
220#41—971no—
221#41—971no—
222#41—1941no—
223#41—1941no—
224#41—1941no—
225#41—1941no—
226#41C2×#8760411yescharacteristic
227C2×Q8—971no—
228C2×Q8—971no—
229C2×D4—971no—
230C2×D4—1941no—
231C2×D4—1941no—
232C2×D4C2×#8760411yescharacteristic
233C22×C4—971no—
234C22×C4—1941no—
235C22×C4—1941no—
236C22×C4—1941no—
237Q16—972no—
238Q16—972no—
239SD16—972no—
240SD16—972no—
241SD16—972no—
242SD16C2×#8760412yescomplemented
243D8—972no—
244D8C2×#8760412yescomplemented
245M16—972no—
246M16—971no—
247M16—971no—
248M16—971no—
249M16—972no—
250M16—972no—
251M16—971no—
252M16—971no—
253M16—1942no—
254M16—1942no—
255M16—1942no—
256M16—1942no—
257M16—1941no—
258M16—1941no—
259M16C2×#8760411yescharacteristic
260C2×C8—972no—
261C2×C8—972no—
262C2×C8—1942no—
263C2×C8—1942no—
264C2×C8—1942no—
265C2×C8—1942no—
266C2×C8—1941no—
267C2×C8—1941no—
268C42—971no—
269C42—971no—
270C42—1941no—
271C23—21no—
272C23—971no—
273C23—1941no—
274Q8—971no—
275Q8—972no—
276Q8—1941no—
277Q8—1941no—
278Q8C22×#8760411yescharacteristic
279D4—21no—
280D4—21no—
281D4—971no—
282D4—1941no—
283D4—1941no—
284D4—1941no—
285D4—1942no—
286D4—1942no—
287D4—1941no—
288D4C22×#8760412yes—
289D4C22×#8760411yescharacteristic
290C2×C4—21no—
291C2×C4—971no—
292C2×C4—971no—
293C2×C4—971no—
294C2×C4—971no—
295C2×C4—971no—
296C2×C4—3882no—
297C2×C4—1941no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1942no—
301C2×C4—1942no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4C22×#8760411yescharacteristic
307C8—972no—
308C8—972no—
309C8—972no—
310C8—1941no—
311C8—1941no—
312C8—1942no—
313C8C22×#8760412yes—
314C22—21no—
315C22—22no—
316C22—81no—
317C22—7761no—
318C22—971no—
319C22—971no—
320C22—3881no—
321C22—1941no—
322C22D4×#8760411yescharacteristic
323C4—21no—
324C4—971no—
325C4—971no—
326C4—3881no—
327C4—1942no—
328C4—1941no—
329C4—1942no—
330C4D4×#8760411yescharacteristic
331C4C23×#8760411yescharacteristic
332C2—21no—
333C2—42no—
334C2—41no—
335C2—3881no—
336C2—1941no—
337C2—1941no—
338C2C2×D4×#8760411yescharacteristic
subgroupG/H
centerC2C2×D4×#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
FittingAGL(1,ℤ/8)×C97C4
socleC2×C97C2×C4×D4
prime radical#20261531C2
cosocleC4×D97C24
hypercenterAGL(1,ℤ/8)#87604
hypocenterC97#2956
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20261814C2semidirect
2. #20261813C2semidirect
3. #20261782C2semidirect
4. #20261780C2semidirect
5. #20261778C2semidirect
6. #20261777C2semidirect
7. #20261750C2semidirect
8. D97⋊#104C2semidirect
9. C2×#15690874C2semidirect
10. M16×#87604C2semidirect
12. #15690898C22semidirect
16. #15690897C22semidirect
17. #15690897C22semidirect
18. #15690897C22semidirect
26. #15690881C22semidirect
27. #15690880C22semidirect
28. #15690875C22semidirect
29. #15690875C22semidirect
30. #15690874C22semidirect
31. #15690874C22semidirect
32. #15690874C22semidirect
33. #15690874C22semidirect
34. #15690864C22semidirect
35. C8×#87604C22semidirect
38. #15690843C4semidirect
39. #15690840C4semidirect
40. #15690839C4semidirect
41. #15690836C4semidirect
42. #15690834C4semidirect
46. AGL(1,ℤ/8)×C97C4semidirect
87. Dic388C2×C4semidirect
88. #13095501C2×C4semidirect
89. #13095500C2×C4semidirect
90. #13095499C2×C4semidirect
91. #13095499C2×C4semidirect
92. #13095496C2×C4semidirect
98. SD16×C97C2×C4semidirect
99. D8×C97C2×C4semidirect
170. C97#2956semidirect
189. AGL(1,ℤ/8)#87604semidirect
242. SD16C2×#87604semidirect
244. D8C2×#87604semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202618142
#202618132
#202617822
#202617802
#202617782
#202617772
#202617502
D97⋊#1042
C2×#156908742
M16×#876042
#202615312
#295697